On the rate of convergence in periodic homogenization of
scalar first-order ordinary differential equations
H. Ibrahim
∗,†
, R. Monneau
∗
October 22, 2008
Abstract
In this paper, we study the rate of convergence in periodic homogenization of scalar ordinary differential equations. We
provide a quantitative error estimate between the solutions of a first-order ordinary differential equation with rapidly oscillating coefficients, and the solution of the limiting homogenized equation. As an application of our result, we obtain an error
estimate for the solution of some particular linear transport equations.
AMS subject classifications: 78M40, 49K15, 65L70, 34C35
Key words: Homogenization of ordinary differential equations, error estimates, linear transport equations
1
Introduction
1.1
Homogenization of an ODE
In this paper, we consider the solutions of the following first-order ordinary differential equation:

ǫ

 uǫ = f u , t , uǫ , t , t > 0,
t
ǫ ǫ

 uǫ (0) = u ,
(1.1)
0
ǫ
ǫ
where ǫ > 0, uǫt stands for du
dt or equivalently ∂t u , and u0 is a real number. We are interested in the rate of
ǫ
convergence of the solution u to its limit in the framework of periodic homogenization. We employ the following
assumptions on the function f :
• (A1) Regularity: the function f : R4 → R is a bounded Lipschitz continuous function with α = Lip(f ) its
Lipschitz constant, and β = kf kL∞ (R4 ) ;
• (A2) Periodicity: for any (v, τ, u, t) ∈ R4 , we have:
f (v + l, τ + k, u, t) = f (v, τ, u, t) for any (l, k) ∈ Z2 ;
• (A3) Monotonicity: for any (v, τ, t) ∈ R3 ,
the function u 7−→ f (v, τ, u, t) is non-increasing.
Let us make short comments on these assumptions. Remark that assumption (A1) ensures the existence and
uniqueness of the solution uǫ of (1.1) via the Cauchy-Lipschitz theorem. Moreover, the assumed boundedness of f
is not a restrictive condition while we work on any finite time interval [0, T ]. The monotonicity assumption (A3)
∗ Université Paris-Est, Ecole des Ponts, CERMICS, 6 et 8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-laVallée Cedex 2, France. E-mails: [email protected], [email protected]
† CEREMADE, Université Paris-Dauphine, Place De Lattre de Tassigny, 75775 Paris Cedex 16, France
1
may seem unnecessary at a first glance, but will be indeed useful to guarantee the uniqueness of the solution to the
homogenized equation (see Proposition 1.4). Moreover, assumption (A3) will play a crucial role to establish the
rate of convergence of uǫ to its limit u0 (see for instance Section 4).
In order to define the homogenized equation, we will use the following proposition:
Proposition 1.1 (Definition and properties of the effective slope f ). Fix (u, t) ∈ R2 . Then there exists
λ ∈ R such that for any initial data u0 ∈ R, the solution v ∈ C 1 ([0, ∞); R) of the following ordinary differential
equation:
(
vτ = f (v, τ, u, t), τ > 0,
v(0) = u0 ,
satisfies
v(τ )
→λ
τ
as
τ → ∞.
(1.2)
Let us set the effective slope:
f (u, t) = λ.
(1.3)
Then the following holds:
(
f : R2 → R is continuous.
(1.4)
For any t ≥ 0, the map u 7→ f (u, t) is non-increasing.
Let us mention that, for some specific functions f , explicit formulas for f can be obtained (see for instance [23],
and the examples below).
Example 1.2 For f (v, τ, u, t) = −u + cos(2πv), we have f (u, t) =
√
f (u, t) ∼ −c u − 1
as
R
1
dv
0 −u+cos(2πv)
u → 1+ ,
−1
for u > 1, and
for some constant c > 0.
Example 1.3 For f (v, τ, u, t) = −u + | sin(2πv)|, we have, for some constant c > 0:
f (u, t) ∼
c
| log |u||
as
u → 0− .
(1.5)
Example 1.2 shows in particular that even for analytic f , the function f could be non-Lipschitz. Example 1.3 shows
a case where f is not Hölder continuous when f is Lipschitz continuous. The proof of Example 1.3 will be given in
the Appendix. At this stage, we can write the homogenized equation associated to equation (1.1) as follows:
(
u0t = f (u0 , t), t > 0,
(1.6)
u0 (0) = u0 .
Even if f may not be Lipschitz continuous in u, we can show the existence and uniqueness of the solution of (1.6),
taking advantage of the monotonicity of f (u0 , t) in u0 . Indeed, we have:
Proposition 1.4 (Existence and uniqueness). Under assumption (1.4) on f , there exists a unique solution
u0 ∈ C 1 ([0, ∞); R) of (1.6).
It is worth noticing that assumption (1.4) satisfied by f in the homogenized equation is our motivation to make
assumption (A3) on f . We can now state our main result:
2
Theorem 1.5 (Error estimate for ODEs). Under assumptions (A1)-(A2)-(A3), if uǫ is the solution of (1.1),
and u0 is the solution of the homogenized equation (1.6), then for every C > 0, ǫ > 0, and every T ≥ Cǫ| log ǫ|, we
have the following estimate:
cT
kuǫ − u0 kL∞ (0,T ) ≤
,
(1.7)
| log ǫ|
where c > 0 is a positive constant only depending on C and on α, β defined in assumption (A1).
Such a result for a general monotone system of ODEs seems to be completely open. The above estimate in
1
| log ǫ|
is in fact related to the behavior of f in Example 1.3, which is the worst possible regularity of f . Moreover, it is
possible to show that under the condition T ≥ Cǫ| log ǫ|, inequality (1.7) is sharp, see the following example whose
proof will be given in the Appendix:
Example 1.6 Let f (v, τ, u, t) = g(v + τ ) − 1 with a 1-periodic function g satisfying
g(w) = |w − 1/2| for
w ∈ [0, 1].
(1.8)
In this case f (u, t) = −1. Let us choose the initial data u0 = 0. Then for any δ > 0, we have the following estimate
between the solution uǫ to (1.1) and u0 to (1.6):
uǫ (t) − u0 (t) ∼
t
2δ| log ǫ|
for
t = δǫ| log ǫ|.
Remark 1.7 It is worth mentioning that assumption (A1) could be replaced by the weaker assumption:
• (A1)’ Regularity: the function f : R4 → R is a bounded continuous function such that for every τ ∈ R, the
function f (., τ, ., .) is Lipschitz continuous.
1.2
Application to the homogenization of linear transport equations
For x = (x1 , x2 ) ∈ R2 , let us consider a vector field aǫ = (aǫ1 , aǫ2 ) defined as follows

 aǫ (x1 , x2 ) = −f x1 , x2 , x1 , x2
1
ǫ ǫ
 aǫ (x , x ) = 1,
2
1
(1.9)
2
with a function f satisfying (A1)-(A2)-(A3). We consider the viscosity solution V ǫ (t, x) of the following linear
transport equation:
( ǫ
Vt + aǫ · ∇V ǫ = 0
on (0, ∞) × R2
(1.10)
V ǫ (0, x) = V0 (x)
on R2 ,
where V0 : R2 → R is a Lipschitz continuous function. The existence and uniqueness of a viscosity solution V ǫ
of (1.10) is ensured since aǫ ∈ W 1,∞ (R2 ) and V0 is Lipschitz continuous (see for instance [3]). The expected
homogenized equation associated to (1.10) is:
( 0
Vt + a · ∇V 0 = 0
on (0, ∞) × R2
(1.11)
V 0 (0, x) = V0 (x)
on R2 ,
with the vector field a = (a1 , a2 ) defined as:
(
a1 (x1 , x2 ) = −f (x1 , x2 ),
with f given by (1.3).
a2 (x1 , x2 ) = 1.
As a consequence of Theorem 1.5, we will show in Section 6 the following result:
3
(1.12)
Theorem 1.8 (Error estimate for linear transport equations). Under the previous assumptions, there exists
a Lipschitz continuous function V 0 which is a viscosity solution of (1.11), such that for any C > 0, ǫ > 0, and
T ≥ Cǫ| log ǫ|, the solution V ǫ of (1.10) satisfies:
kV ǫ − V 0 kL∞ (R2 ×(0,T )) ≤
c′ T
| log ǫ|
(1.13)
where c′ = cLip(V0 ), and c > 0 is the constant given in Theorem 1.5.
Choosing the initial condition V0 (x) = x1 , we can easily deduce from Example 1.6 that inequality (1.13) is also
sharp for T ≥ Cǫ| log ǫ|. Here, in this application, the vector field aǫ is quite special. The interested reader could be
referred to [14] for some other examples of vector fields in 2D, where a homogenization result is presented without
any rate of convergence. In [26], the author gives some non-explicit error estimates for linear transport equations in
the particular case of periodic vector field aǫ . However, these error estimates obtained in [26] may depend strongly
on the irrationality of the rotation number ω0 associated to the vector field aǫ (where ω0 is nothing else than −f in
our application). On the contrary, estimate (1.13) only depends on some bounds of the data of the problem, and
are completely uniform with respect to the rotation number.
Remark that when f (v, τ, u, t) is independent of u and t, we have a much better estimate:
Theorem 1.9 (Better error estimate). Under the assumptions of Theorem 1.8, if f (v, τ, u, t) is independent
of u and t, then we have:
kV ǫ − V 0 kL∞ (R2 ×(0,T )) ≤ c′′ ǫ, ∀T ≥ 0,
where c′′ = ξLip(V0 ), with ξ (given in Proposition 2.1) only depends on β defined in assumption (A1).
The proof of Theorem 1.9 will also be given in Section 6.
1.3
Brief review of the literature
The pioneering work (via the theory of viscosity solutions) to periodic homogenization was established in [17].
Starting from [17], the homogenization theory for Hamilton-Jacobi equations has received a considerable interest.
There is a huge literature that we cannot cite in details, but the interested reader can for instance see [1, 5, 4, 11,
18, 15, 16] and the references therein. Another aspect concerning homogenization of SDEs (stochastic differential
equations) has also been studied by several authors (see for instance [20, 13, 6, 24]). These problems are related to
our problem when the SDE reduces to an ODE.
To our knowledge, the question of estimating the rate of convergence in homogenization of PDEs has not been
widely tackled up elsewhere in the literature. We can cite [7] for several error estimates concerning the rate of
convergence of the approximation scheme to the effective Hamiltonian. We can also cite the work in [8] about the
rate of convergence in periodic homogenization of first-order stationary Hamilton-Jacobi equations, where an error
estimate in ǫ1/3 is obtained for Hamilton-Jacobi equations with Lipschitz effective Hamiltonian.
For the problems of homogenization of ODEs, we refer the reader to [23]. We also refer the reader to [2, 9, 10,
19, 21, 22, 25] for problems on homogenization of nonlinear first-order ODEs and/or the associated linear transport
equations. As mentioned above, we refer the reader to [26] for some other error estimates for linear transport
equations.
1.4
Organization of the paper
The paper is organized as follows. In Section 2, we present the proof of an ergodicity result (Proposition 2.1) that
defines f = λ. We also present the proofs of Propositions 1.1 and 1.4. In Section 3, we give a result of stability of
λ under additive perturbation (Proposition 3.1). A basic error estimate (Proposition 4.1) is presented in Section 4.
Section 5 is devoted to show our main result of estimating the rate of convergence (Theorem 1.5). In Section 6, we
give an application to the case of linear transport equations (Theorems 1.8 and 1.9). We end up in Section 7 with
an Appendix where we give the proof of Examples 1.3 and 1.6.
4
2
Ergodicity and preliminary facts
In this section we present the proof of Propositions 1.1 and 1.4. We first start with the following ergodicity result
which is a particular case of [12, Proposition 4.2]. However, we give the proof in our particular case for the sake of
completeness.
Proposition 2.1 (Ergodicity). Let g(v, τ ) : R2 → R be a function satisfying:
• (H1) Regularity: g is Lipschitz continuous with kgkL∞ (R2 ) ≤ β;
• (H2) Periodicity: g(v + l, τ + k) = g(v, τ ) for any (l, k) ∈ Z2 , (v, τ ) ∈ R2 .
Let v be the solution of the following equation
(
vτ = g(v, τ ),
τ > 0,
(2.1)
v(0) = v0 ,
then there exist a constant λ ∈ R (independent of the initial data v0 ) such that for every τ, τ ′ ≥ 0, we have:
|v(τ ) − v(τ ′ ) − λ(τ − τ ′ )| ≤ ξ
with
ξ = 1 + 2β.
(2.2)
Remark 2.2 Under our assumptions, it is possible (see [12, Theorem 1.5]) to show the existence of a hull function
h : R × R → R satisfying:


 h(τ + 1, x) = h(τ, x)
h(τ, x + 1) = h(τ, x) + 1


hx ≥ 0,
such that U (τ, x) = h(τ, x + λτ ), with λ given in Proposition 2.1, is a viscosity solution of:
(
Uτ = g(U, τ )
U (τ = 0, x) = x.
This function h may be discontinuous, but its existence suggests that
v(τ ) = h(τ, λτ )
(2.3)
is formally a classical of (2.1). The expression (2.3) allows to understand estimate (2.2) and also suggests that
v(τ ) − λτ could be quasi-periodic in some cases. We emphasize the fact that in the proof of Proposition 2.1, we do
not use any notion of hull function, but propose a completely independent proof.
Proof of Proposition 2.1. For any T > 0, define the two quantities:
λ+ (T ) = sup
τ ≥0
v(τ + T ) − v(τ )
T
and λ− (T ) = inf
τ ≥0
v(τ + T ) − v(τ )
.
T
These quantities are finite since vτ = g is bounded. The proof is divided into three steps.
Step 1: Estimate of |λ+ − λ− |.
Let δ > 0 be an arbitrary constant. From the definition of λ± (T ), there always exists τ ± such that
±
± ±
λ (T ) − v(τ + T ) − v(τ ) ≤ δ.
T
(2.4)
Denote by ⌊t⌋ and ⌈t⌉ as the floor and the ceil integer parts of the real number t respectively. We consider
k = ⌊τ − − τ + ⌋,
τe− = τ − − k
5
and l = ⌈v(e
τ − ) − v(τ − )⌉,
(2.5)
and consider w(τ ) = v(τ + k) + l. Using (2.1) and (H2), we check that w is a solution of the following equation
(
wτ = g(w, τ ), τ > 0
(2.6)
w(e
τ − ) = v(τ − ) + l.
We remark, from (2.5), that
τ + ≤ τe− < τ + + 1,
and
(2.7)
v(e
τ − ) ≤ w(e
τ − ) < v(e
τ − ) + 1.
(2.8)
From (2.1), (2.6) and (2.8), the comparison principle for ODEs gives in particular
for all τ ≥ τe− .
v(τ ) ≤ w(τ )
(2.9)
If we suppose T ≥ 1, we obtain from (2.7) that τ + + T ≥ τe− and hence (using (2.9)):
v(τ + + T ) ≤ w(τ + + T ).
(2.10)
Direct computations give
(v(τ − + T ) − v(τ − )) − (v(τ + + T ) − v(τ + ))
= (w(e
τ − + T ) − w(τ + + T )) + (v(e
τ − ) − w(e
τ − ))
+(w(τ + + T ) − v(τ + + T )) + (v(τ + ) − v(e
τ − )),
where, from (2.8) and (2.10), we deduce that (v(τ − + T ) − v(τ − )) − (v(τ + + T ) − v(τ + )) ≥ −1 − 2kgk∞. This
∞
+ 2δ, and since this is true for any δ > 0,
inequality, together with (2.4) show that 0 ≤ λ+ (T ) − λ− (T ) ≤ 1+2kgk
T
we obtain
1 + 2kgk∞
|λ+ (T ) − λ− (T )| ≤
.
T
However, in the case where T ≤ 1, we always have
|v(τ +T )−v(τ )|
T
0 ≤ λ+ (T ) − λ− (T ) ≤
then
ξ
T
|λ+ (T ) − λ− (T )| ≤
≤ kgk∞ ≤
kgk∞
T
and therefore
2kgk∞
,
T
for every T > 0.
(2.11)
Step 2: Existence of the limit of λ± (T) as T → ∞.
First, if we compute λ+ (P T ) for P ∈ N \ {0}, we get:
" P
#
1 X v(τ + iT ) − v(τ + (i − 1)T )
+
λ (P T ) = sup
≤ λ+ (T ).
T
τ >0 P
i=1
Similarly, we get λ− (P T ) ≥ λ− (T ). Consider T1 , T2 > 0 such that T1 P = T2 Q for some P, Q ∈ N \ {0}. Using this
and (2.11), we have
λ+ (T2 ) ≥ λ+ (T2 Q) = λ+ (T1 P ) ≥ λ− (T1 P ) ≥ λ− (T1 ) ≥ λ+ (T1 ) −
similarly we have λ+ (T2 ) − λ+ (T1 ) ≤
ξ
T2 ,
ξ
,
T1
then
|λ+ (T1 ) − λ+ (T2 )| ≤ max
6
ξ ξ
,
T1 T2
.
(2.12)
By the same arguments as above we can get
|λ− (T1 ) − λ− (T2 )| ≤ max
ξ ξ
,
T1 T2
.
(2.13)
Recall that (2.12) and (2.13) are true when T1 /T2 is rational. By an approximation argument, joint with the
continuity of λ± , it is easy to see that this is still true when T1 /T2 is any positive real number. Moreover, the
identities (2.12) and (2.13) give that the sequence (λ± (T ))T is a Cauchy sequence as T → ∞, and hence it has a
limit:
lim λ± (T ) = λ,
(2.14)
T →∞
which is the same limit because of (2.11). From (2.14), (2.12) and (2.13), inequality (2.2) directly follows.
Step 3: Independence of v0 .
The fact that λ is independent of v0 follows directly from the comparison principle and inequality (2.2).
2
We can now present the proof of Propositions 1.1 and 1.4.
Proof of Proposition 1.1. Using inequality (2.2) of Proposition 2.1, we can easily see that (1.2) directly follows.
It remains to show (1.4). We argue in two steps.
Step 1: Monotonicity of f .
Let u ≤ ũ. Call λ = f (u, t) and λ̃ = f (ũ, t). Let v and ṽ be the solutions of:
(
vτ = f (v, τ, u, t), τ > 0,
v(0) = u0 ,
and
(
ṽτ = f (ṽ, τ, ũ, t),
ṽ(0) = u0 ,
τ > 0,
respectively. Assume without loss of generality that u0 = 0. Using (A3), we deduce that f (ṽ, τ, ũ, t) ≤ f (ṽ, τ, u, t).
Hence, the comparison principle gives:
ṽ(τ ) ≤ v(τ )
for every τ ≥ 0.
(2.15)
From inequality (2.2) of Proposition 2.1, we have:
|v(τ ) − λτ | ≤ ξ
and |ṽ(τ ) − λ̃τ | ≤ ξ
for all τ ≥ 0.
(2.16)
We then easily conclude that λ̃ ≤ λ as a consequence of (2.15) and (2.16).
Step 2: Continuity of f .
We refer the reader to Proposition 3.2 which implies in particular the continuity of f . The main idea of the proof
2
is to apply a perturbation argument using the inequality |λ± (T ) − λ| ≤ Tξ .
Proof of Proposition 1.4.
Global existence. This is a direct consequence of the Cauchy-Péano theorem, using in particular the continuity of
f (see 1.4).
7
Uniqueness. Assume that there exists u1 ∈ C 1 ([0, ∞); R) another solution of (1.6). Define k(t) = |u0 (t) − u1 (t)|,
we compute (with the sign function sgn(x) = x/|x| if x 6= 0):
kt (t)
= (u0t (t) − u1t (t)) sgn(u0 (t) − u1 (t))
= (f (u0 (t), t) − f (u1 (t), t)) sgn(u0 (t) − u1 (t)) ≤ 0,
where for the last line we have used the monotonicity of f (see (1.4)). This immediately implies that u0 = u1 .
3
2
A stability result for the effective slope f
In this section, we will show a stability result for the term λ given by Proposition 2.1 under a perturbation of (2.1)
of the form:
(
vt = gγ (v, t) = g(v, t) ± γ, t > 0
(3.1)
v(0) = v0 ,
where γ > 0 is a small real number. More precisely, we have the following proposition:
Proposition 3.1 (Stability result). Take 0 < γ < 1. Let λγ be the effective slope given by Proposition 2.1, which
is associated to equation (3.1), and let λ0 be the one corresponds to γ = 0 in (3.1). Then we have the following
estimate:
ξ̄
with ξ̄ = (3 + 2ξ)(1 + 2L),
(3.2)
|λγ − λ0 | ≤
| log γ|
∂g and L = ∂v
∞ 2 .
L
(R )
Proof. We assume, for the sake of simplicity, that gγ = g + γ. In this case, it is easy to check that λγ ≥ λ0 . The
other case with gγ = g − γ is treated similarly. We first transform our ODE problem into a PDE one by setting v γ
as the solution of the following equation:
( γ
vt (t, x) = g(v γ (t, x), t) + γ, in (0, ∞) × R
(3.3)
v γ (0, x) = x,
x ∈ R.
The proof is divided into three steps.
Step 1: A control on vxγ .
Using comparison principle arguments for (3.3), it is easily checked that v γ (t, .) is a non-decreasing function satisfying v γ (t, x + 1) = v γ (t, x) + 1. We want to control vxγ (t, .) for any t. For this reason, we proceed as follows. Define
for z ≥ 0:
η(t, x) = v γ (t, x + z) − zeLt , t > 0, x ∈ R.
We compute
ηt (t, x)
= vtγ (t, x + z) − zLeLt
= g(η(t, x) + zeLt , t) − zLeLt + γ
≤ g(η(t, x), t) + γ,
which proves that η is a sub-solution of (3.3) with η(0, x) = v γ (0, x + z) − z = v γ (0, x), and therefore, by the
comparison principle, we obtain
η(t, x) = v γ (t, x + z) − zeLt ≤ v γ (t, x)
hence for any t ≥ 0, we have 0 ≤ v γ (t, x + z) − v γ (t, x) ≤ zeLt , then v γ (t, x) is Lipschitz continuous in the variable
x, satisfying:
0 ≤ vxγ (t, x) ≤ eLt for t ≥ 0 and a.e. x ∈ R.
(3.4)
8
In a similar way, we can obtain a positive bound from below on vxγ , and finally get
e−Lt ≤ vxγ (t, x) ≤ eLt .
(3.5)
Step 2: An upper bound of v γ .
We seek to find an upper bound of v γ by constructing an explicit super-solution of (3.3) with suitable initial data,
and comparing it with v γ . For this purpose, let
w(t, x) = v 0 (t, x + c1 γt),
(t, x) ∈ (0, ∞) × R,
(3.6)
where v0 is equal to v γ for γ = 0, and c1 is positive constant to be precised later. We calculate:
wt (t, x)
=
vt0 (t, x + c1 γt) + c1 γvx0 (t, x + c1 γt)
=
g(w(t, x), t) + c1 γvx0 (t, x + c1 γt),
where from (3.5), we deduce that
LT
wt (t, x) ≥ g(w(t, x), t) + c1 γe−Lt .
(3.7)
w(t, x) ≥ v γ (t, x) ∀t ∈ [0, T ], x ∈ R.
(3.8)
Take c1 = e
for some fixed T > 0. Then using (3.7), we get wt (t, x) ≥ g(w(t, x), t) + γ for any t ∈ [0, T ]. Hence
w is a super-solution of (3.3) over [0, T ] whose initial condition w(0, x) = v γ (0, x), which finally gives:
Step 3: Conclusion.
We will now show the error estimate (3.2). To this end, we will estimate both sides of inequality (3.8) involving λ0
and λγ . Firstly, using (2.2) and (3.5), we compute:
|v 0 (t, x + eLT γt) − v 0 (0, x)|
≤ |v 0 (t, x + eLT γt) − v 0 (t, x)| + |v 0 (t, x) − v 0 (0, x)|
≤ eLt eLT γt + λ0 t + ξ.
We take this inequality for t = T and x = 0, we get
w(T, 0) = v 0 (T, eLT γT ) ≤ γT e2LT + λ0 T + ξ.
(3.9)
Secondly, using similar arguments, and the fact that γ < 1, we obtain |v γ (T, 0) − v γ (0, 0) − λγ T | ≤ 2 + ξ, hence
v γ (T, 0) ≥ λγ T − (2 + ξ).
(3.10)
Combining (3.8), (3.9) and (3.10), it follows that
(λγ − λ0 )T ≤ γT e2LT + 2(1 + ξ)
(3.11)
Using (3.11), we deduce that:
|λγ − λ0 | ≤ γe2LT +
2(1 + ξ)
.
T
Since the variable T was arbitrary chosen, let T satisfies γT e2LT = 1 and therefore T ≥
result directly follows.
(3.12)
| log γ|
1+2L .
From (3.12), the
2
An immediate consequence of Proposition 3.1 is the following:
Proposition 3.2 (Modulus of continuity of f ). The function f (u, t) given by (1.3) satisfies for any (u, t) ∈ R2 ,
and for all |v| + |s| < α1 :
ξ̄
,
(3.13)
|f (u + v, t + s) − f (u, t)| ≤
| log α(|v| + |s|)|
where α is given in assumption (A1).
Remark that estimate (3.13) is optimal in view of Example 1.3.
9
4
Basic error estimate
We start this section by considering a discrete scheme associated to the ODE (1.6). Namely, for a given v 0 (which
may be chosen equal to u0 or may be different), and for a time step ∆t > 0, we define the sequence (v k )k∈N as
follows:
v k+1 = v k + λk ∆t, λk = f (v k , k∆t), k ∈ N.
(4.1)
In this section we give a local error estimate between the solution uǫ of (1.1) and the sequence v k . To be more
precise, we will show the following proposition:
Proposition 4.1 (Basic error estimate). Under assumptions (A1), (A2) and (A3) on the function f , let
∆t > 0 be small enough (depending only on α and β), and take
ei = |uǫ (i∆t) − v i |,
Then we have:
e1 ≤ e0 + cǫ +
where c = c(α, β) > 0 is a positive constant.
i = 0, 1.
c∆t
,
| log c∆t|
(4.2)
The proof of the above proposition will be presented later in this section. In what follows in this section and in
Section 5, we will assume that the arguments of the various logarithms are all less than 1.
Lemma 4.2 (Refined basic error estimate). Under the same hypothesis of Proposition 4.1, let
ǫ
i
e+
i = max(0, u (i∆t) − v ),
−
ǫ
0
and let d+
1 = sup (max(0, u (t) − v )), d1 =
t∈[0,∆t]
ǫ
i
e−
i = min(0, u (i∆t) − v ),
i = 0, 1
inf (min(0, uǫ (t) − v 0 )). Then we have:
t∈[0,∆t]
+
e+
1 ≤ e0 + (2 + ξ)ǫ +
ξ̄∆t
| log α(|d−
1 | + ∆t)|
(4.3)
−
e−
1 ≥ e0 − (2 + ξ)ǫ −
¯
ξ∆t
,
| log α(d+
1 + ∆t)|
(4.4)
and
with ξ = 1 + 2β and β defined in (A1).
Proof. In order to get estimates (4.3), (4.4), the main idea is to freeze the
last two arguments of f , and to use
ǫ
some comparison arguments. We start by estimating the term f uǫ , ǫt , uǫ , t from below. Since uǫ (t) ≤ v 0 + d+
1 for
t ∈ [0, ∆t], we deduce, using (A3), that
ǫ
ǫ
ǫ
ǫ
u t ǫ
u t 0
u t 0
u t 0
+
f
, ,u ,t − f
, ,v ,0 ≥ f
, , v + d1 , t − f
, ,v ,0 ,
ǫ ǫ
ǫ ǫ
ǫ ǫ
ǫ ǫ
and hence, from (A1), we get
f
uǫ t ǫ
, ,u ,t
ǫ ǫ
≥f
uǫ t 0
, , v , 0 − α(d+
1 + ∆t).
ǫ ǫ
Using similar arguments, we can also show
ǫ
ǫ
u t 0
u t ǫ
, ,u ,t ≤ f
, , v , 0 + α(|d−
f
1 | + ∆t).
ǫ ǫ
ǫ ǫ
(4.5)
(4.6)
−
We know from the definition of e+
0 and e0 that:
+
ǫ
0
v 0 + e−
0 ≤ u (0) ≤ v + e0 .
10
(4.7)
Let w ǫ and wǫ be the solutions of the following ODEs:

ǫ

 wǫ = f w , t , v 0 , 0 − α(d+ + ∆t),
t
1
ǫ ǫ

 wǫ (0) = v 0 + e−
t>0
(4.8)
0
and

ǫ

 w ǫ = f w , t , v 0 , 0 + α(|d− | + ∆t),
t
1
ǫ ǫ

 w ǫ (0) = v 0 + e+ ,
0
t>0
(4.9)
respectively. From (4.5), (4.6) and (4.7), we deduce, using the comparison principle, that:
w ǫ ≤ uǫ ≤ w ǫ
ǫ
on [0, ∆t].
(4.10)
ǫ
Applying Proposition 2.1 to the functions w and w , we know that there exists two real numbers λ0 and λ0 such
−
that for t ∈ [0, ∆t] (assuming α(d+
1 + ∆t) ≤ 1 and α(|d1 | + ∆t) ≤ 1):
|w ǫ (t) − wǫ (0) − λ0 t| ≤ (2 + ξ)ǫ
and |w ǫ (t) − w ǫ (0) − λ0 t| ≤ (2 + ξ)ǫ.
(4.11)
Inequalities (4.10) and (4.11) give:
+
ǫ
0
v 0 + e−
0 + λ0 t − (2 + ξ)ǫ ≤ u (t) ≤ v + e0 + λ0 t + (2 + ξ)ǫ,
0
t ∈ [0, ∆t].
(4.12)
Using Proposition 3.1, we obtain, for λ0 = f (v , 0), that
0 ≥ λ0 − λ0 ≥ −
and
ξ¯
| log α(d+
1 + ∆t)|
ξ¯
.
| log α(|d−
1 | + ∆t)|
The above two inequalities, together with (4.7) and (4.12) give the result.
0 ≤ λ0 − λ0 ≤
(4.13)
(4.14)
2
Two immediate corollaries of the above lemma are the following:
Corollary 4.3 (Refined estimates involving e±
i ). Under the same hypothesis of Lemma 4.2, we have:
+
e+
1 ≤ e0 + (2 + ξ)ǫ +
ξ̄∆t
| log α1 (|e−
0 | + ∆t)|
(4.15)
−
e−
1 ≥ e0 − (2 + ξ)ǫ −
¯
ξ∆t
,
| log α1 (e+
0 + ∆t)|
(4.16)
and
where α1 = α1 (α, β) > 0 is a positive constant.
+
−
−
Proof. We have d+
1 ≤ e0 + β∆t and |d1 | ≤ |e0 | + β∆t (recall that β = kf kL∞ (R4 ) ). Therefore, the result can be
deduced from (4.3) and (4.4).
2
Corollary 4.4 (Refined estimates with continuous time). Under the same hypothesis of Proposition 4.1, for
every t ∈ [0, ∆t], define the continuous function ē by:
ē(t) = uǫ (t) − (v 0 + λ0 t).
(4.17)
Also define e+ (t) = max(0, ē(t)) and e− (t) = min(0, ē(t)). Then we have for 0 ≤ t1 < t2 ≤ ∆t and c = c(α, β) > 0:
e+ (t2 ) ≤ e+ (t1 ) + cǫ +
and
e− (t2 ) ≥ e− (t1 ) − cǫ −
c(t2 − t1 )
| log c(|e− (t1 )| + (t2 − t1 ))|
(4.18)
c(t2 − t1 )
.
1 ) + (t2 − t1 ))|
(4.19)
| log c(e+ (t
11
Proof. We apply the same proof (word by word) of Lemma 4.2 and Corollary 4.3, with the origin 0 shifted to t1
and the time step ∆t replaced by t2 − t1 .
2
Now we are ready to prove Proposition 4.1.
Proof of Proposition 4.1. From the definition (4.17) of ē, we know that
e0 = |ē(0)| and e1 = |ē(∆t)|.
The case where e1 = ē(∆t) = 0 is obvious. Four cases could be considered.
+
−
Case 1. (ē(0) ≥ 0 and ē(∆t) > 0). In this case we have e0 = e+
0 , e1 = e1 and e0 = 0. Therefore (4.2) is an
immediate consequence of (4.15).
Case 2. (ē(0) ≤ 0 and ē(∆t) < 0). Similar to Case 1.
Case 3. (ē(0) < 0 and ē(∆t) > 0). In this case e1 = e+ (∆t). Let the time t−+ be defined as follows:
t−+ = max{t ∈ (0, ∆t);
ē(t) = 0}.
Using inequality (4.18) with t1 = t−+ , t2 = ∆t and ∆t = ∆t − t−+ ≤ ∆t, we get:
e1 ≤ cǫ +
c∆t
c∆t
≤ e0 + cǫ +
| log c∆t|
| log c∆t|
where we have used the fact that e+ (t1 ) = e− (t1 ) = 0, and hence (4.2) follows.
Case 4. (ē(0) > 0 and ē(∆t) < 0). Similar to Case 3.
5
2
Estimate of the rate of convergence
This section is entirely devoted to the proof of Theorem 1.5. Let T > 0 and let ∆t > 0 be such that
n∆t = T,
n ∈ N,
(5.1)
where n to be chosen large enough (the choice of ∆t will be given later). In order to estimate kuǫ − u0 kL∞ (0,T ) , we
add and subtract v, the continuous piecewise linear function passing through the points (k∆t, v k ), k = 0 · · · n. In
other words
v(t) = v k + (t − k∆t)λk , k∆t ≤ t ≤ (k + 1)∆t, k = 0 · · · n − 1,
where v k and λk are defined in (4.1). We start by stating the following corollary that generalizes Proposition 4.1.
Corollary 5.1 (Basic error estimate in continuous time). Under assumptions (A1), (A2) and (A3), let
T > 0, and ∆t > 0 given by (5.1). Define the function e(t) by:
e(t) = |uǫ (t) − v(t)|,
t ∈ [0, T ].
(5.2)
For k = 0 · · · n − 1, call ek = e(k∆t). Then for k∆t ≤ t ≤ (k + 1)∆t, we have:
e(t) ≤ ek + cǫ +
c∆t
,
| log c∆t|
where c = c(α, β) > 0 is a positive constant (the same given by Proposition 4.1).
12
(5.3)
Proof. We simply apply Proposition 4.1 with the origin 0 shifted to k∆t and with the time step ∆t replaced by
δt = t − k∆t ≤ ∆t.
2
At this stage, we can show an inequality similar to (5.3) with e(t) and ek replaced respectively by the functions
e0 (t) = |u0 (t) − v(t)|
and e0k = e0 (k∆t),
(5.4)
where u0 is the solution of (1.6). Indeed, we have the following proposition:
Proposition 5.2 (Basic error estimate for the homogenized ODE). Let f be the function given by (1.3),
and enjoying the properties given by Propositions 1.1 and 3.2. Let T > 0, and ∆t > 0 given by (5.1). Then for
k∆t ≤ t ≤ (k + 1)∆t, k = 0 · · · n − 1, we have:
c∆t
| log c∆t|
e0 (t) ≤ e0k +
(5.5)
where e0 , e0k are given by (5.4), and c = c(α, β) > 0 is a positive constant.
Sketch of the proof. Although the proof is an adaptation of the proof of inequality (5.3) (it suffices to deal with
f (u, t) instead of f (v, τ, u, t), and to take ǫ = 0), we will indicate the main points where it slightly differs. The
crucial idea is that, on the one hand, the monotonicity of the function f (v, τ, u, t) with respect to the variable u
(see assumption (A3)) is replaced by the monotonicity of f (u, t) with respect to u (see (1.4)). On the other hand,
the fact that f is Lipschitz continuous (see assumption (A1)) is replaced by the fact that f satisfies a modulus of
continuity (see Proposition 3.2), and the fact that f (u, t) = λ(u, t).
Let us go into the details. In fact, the proof of Lemma 4.2 can be adapted where inequalities (4.5) and (4.6) are
replaced by
f (u0 , t) ≥ f (v 0 , 0) −
ξ̄
,
+ ∆t)|
| log α(d0,+
1
and
f (u0 , t) ≤ f (v 0 , 0) +
ξ¯
| log α(|d0,−
1 |
+ ∆t)|
d0,+
= sup (max(0, u0 (t) − v 0 ))
1
with
(5.6)
t∈[0,∆t]
,
with
d0,−
=
1
inf (min(0, u0 (t) − v 0 ))
t∈[0,∆t]
(5.7)
respectively. Here we have used the monotonicity of f , and inequality (3.13) given by Proposition 3.2. Having (5.6)
and (5.7) in hands, the sub- and super-solution w ǫ , wǫ defined by (4.8) and (4.9) are replaced by w 0 , w 0 solutions
of

ξ̄

 w 0 = f (v 0 , 0) −
t
| log α(d0,+
+ ∆t)|
1

 0
0
0,−
w (0) = v + e
with e0,− = min(0, u0 (0) − v 0 )
and


 w 0 = f (v 0 , 0) +
t
respectively, and we have


0
0
w (0) = v + e
ξ¯
| log α(|d0,−
1 |+
0,+
0,+
with
e
∆t)|
= max(0, u0 (0) − v 0 )
0
w0 (t) − w0 (0) − λ0 t = 0 and w 0 (t) − w 0 (0) − λ00 t = 0
with
ξ̄
0
λ0 = w0t = λ0 −
| log α(d0,+
1
+ ∆t)|
and λ00 = w 0t = λ0 +
(5.8)
ξ̄
| log α(|d0,−
1 |
+ ∆t)|
,
(5.9)
where we recall the reader that λ0 = f (v 0 , 0). Remark that (5.8) replaces (4.11), while (5.9) gives (as a replacement
of (4.13) and (4.14)):
0
0 ≥ λ0 − λ0 = −
ξ̄
| log α(d0,+
1
+ ∆t)|
and 0 ≤ λ00 − λ0 =
13
ξ̄
| log α(|d0,−
1 |
+ ∆t)|
.
At this point, the rest of the proof proceeds in a very similar way as the proof of (5.3) with ǫ = 0, and a possible
changing of the constants but always depending on α and β.
2
Now we are ready to present the proof of our main result.
Proof of Theorem 1.5. We first note that the constant c > 0 may certainly differ from line to line in the proof.
We decompose the quantity |uǫ (t) − u0 (t)| into two pieces:
e0 (t)
e(t)
}|
{
z
}|
{ z
|uǫ (t) − u0 (t)| ≤ |uǫ (t) − v(t)| + |u0 (t) − v(t)| .
(5.10)
In order to estimate e(t), we iterate inequality (5.3) and we finally obtain for uǫ (0) = u0 = v 0 , and t ∈ [0, T ] with
T = n∆t:
cǫT
cT
cn∆t
≤
+
.
e(t) ≤ e0 + cnǫ +
| log c∆t|
∆t
| log c∆t|
The above inequality gives:
e(t) ≤ cT
1
ǫ
+
∆t | log ∆t|
,
(5.11)
and, from inequality (5.5) of Proposition 5.2, we can show in the same way as above that we also have:
e0 (t) ≤
cT
.
| log ∆t|
(5.12)
Choosing particularly ∆t = Cǫ| log ǫ|, we deduce that ǫ ≪ ∆t = Cǫ| log ǫ| ≪ T , and (from (5.11), (5.12)) that:
e(t) ≤
cT
| log ǫ|
and e0 (t) ≤
cT
,
| log ǫ|
(5.13)
with c in (5.13) depending on the choice of C > 0. finally, inequality (1.7) could now be easily deduced from (5.10)
and (5.13).
2
6
Application: error estimate for linear transport equations
In this section, as an application of our previous results on ODEs, we give the proof of some error estimates for the
homogenization of linear transport equations. Namely, we prove Theorems 1.8 and 1.9. We start with Theorem 1.8
keeping the same notations of Subsection 1.2.
Proof of Theorem 1.8. The proof is divided into four steps. The first three steps are devoted to the definition
of the limit solution V 0 , and to prove that it is a viscosity solution. The proof of the error estimate is done in the
last step.
Step 1: Definition of V 0 .
Because the homogenized vector field a is not Lipschitz, we define our solution V 0 to (1.11) in an indirect way using
the characteristics. Precisely, for (t, x) ∈ (0, ∞) × R2 , x = (x1 , x2 ), we define V 0 (t, x) as follows:
V 0 (t, x) = V0 (X 0 (0; t, x))
(6.1)
where the curve X 0 (τ ; t, x) : τ ∈ R → X 0 (τ ; t, x) ∈ R2 ,
X 0 (τ ; t, x) = (X10 (τ ; t, x), X20 (τ ; t, x))
is the solution of the following ODE:
(
∂τ X 0 = a(X 0 )
X 0 (t) = x.
14
(6.2)
(6.3)
We will see below that this solution is unique. For the sake of simplicity of notations, we will omit the dependence
of X 0 on (t, x) and we will simply write
X 0 (τ ; t, x) = X 0 (τ ).
From (6.3) and (1.9), we can easily check that X20 (τ ) = x2 − t + τ , hence using (6.1), we get
V 0 (t, x) = V0 X10 (0), x2 − t
where X10 satisfies:
(
∂τ X10 = −f X10 , x2 − t + τ
X10 (t) = x1 .
(6.4)
In order to show that X10 (0) is uniquely defined, we solve (6.4) backwards, in other words, we let
0
X 1 (τ ) = X10 (t − τ ).
In this case:
where
0
X1
0
satisfies:
0
V 0 (t, x) = V0 (X 1 (t), x2 − t),
(6.5)

 ∂τ X 01 = f X 01 , x2 − τ
(6.6)

0
X 1 (0) = x1 .
0
From Proposition 1.4, the solution X 1 ∈ C 1 ([0, ∞); R) is unique and hence X10 (0) = X 1 (t) is uniquely determined.
Consequently the function V 0 is well defined.
Step 2: V 0 is Lipschitz continuous.
From Step 1, we know that
0
V 0 (t, x1 , x2 ) = V0 (X 1 (t), x2 − t)
0
(6.7)
with X 1 given by (6.6) also depends on x1 and x2 . Let the function Y : R3 → R be defined as follows (with
simplified notation showing the dependence on the variables (t, x1 , x2 )):
0
Y (t, x1 , x2 ) := X 1 (t).
In order to show that V 0 is Lipschitz, it suffices (see (6.7)) to show that Y is Lipschitz. First, it is easily seen from
(6.6) that Y is Lipschitz in time t. The Lipschitz continuity with respect to the variable x1 directly follows from
the monotonicity of f (see (1.4)), and the comparison principle. In order to show the Lipschitz continuity with
respect to x2 , we first give a formal proof by assuming that f is smooth, and then we present the main idea that
permit to make the proof rigorous. Suppose that
f ∈ C ∞ (R2 ; R) with |f (y, s)| ≤ kf k∞ .
Take
then the above two functions satisfy:
and
Yb = ∂x2 Y
(
(
and Ye = ∂t Y,
∂t Yb = (∂y f )Yb + ∂s f
Yb (0, x1 , x2 ) = 0,
∂t Ye = (∂y f )Ye − ∂s f
Ye (0, x1 , x2 ) = f (x1 , x2 ),
15
(6.8)
(6.9)
respectively. Let Y = Yb + Ye , we get (from (6.8) and (6.9)):
(
∂t Y = (∂y f )Y
Y (0, x1 , x2 ) = f (x1 , x2 ),
which gives, because of the monotonicity of f (see (1.4)), that the function t → |Y (t, ., .)| is non-increasing. Hence
|Y (t, x1 , x2 )| ≤ |f (x1 , x2 )| ≤ kf k∞ .
(6.10)
Since Ye = ∂t Y , we know from (6.6) that |Ye | ≤ kf k∞ where we finally obtain (see (6.10)):
|Yb | = |∂x2 Y | ≤ 2kfk∞ ,
which shows that Y is Lipschitz continuous in the x2 variable. In order to make the proof rigorous, it suffices
to consider a regular approximation of the function f (as for example the convolution with a suitable mollifier
sequence) and then to pass to the limit.
Step 3: V 0 is a viscosity solution of (1.11).
For the definition and the study of the theory of viscosity solutions, we refer the reader to [3]. Let us take
φ, φ ∈ C 1 (R3 ; R) such that V 0 − φ (resp. V 0 − φ) has a local maximum (resp. local minimum) at some point
(t, x) ∈ (0, ∞) × R2 (resp. (t, x) ∈ (0, ∞) × R2 ), with V 0 (t, x) = φ(t, x) and V 0 (t, x) = φ(t, x). In order to show
that V 0 is a viscosity solution of (1.11), we need to show the following two inequalities:
∂t φ(t, x) + (a · ∇φ)(t, x) ≤ 0
(6.11)
∂t φ(t, x) + (a · ∇φ)(t, x) ≥ 0.
(6.12)
and
We only show inequality (6.11). In fact, inequality (6.12) can be proved in exactly the same way. For any t ∈ [0, t]
define the function φ by
φ : t → φ(t) = φ(t, X 0 (t; t, x)),
where X 0 is defined in (6.3). Let us show an inequality on φ in the interval [t − r, t] for r > 0 small enough. Remark
that X 0 (t; t, x) = x. Hence, for t ∈ [t − r, t], (t, X 0 (t; t, x)) is close to (t, x) and therefore (since V 0 − φ has a local
maximum at (t, x) with V 0 (t, x) = φ(t, x)) we get:
φ(t) = φ(t, X 0 (t; t, x)) ≥ V 0 (t, X 0 (t; t, x)) = V0 (X 0 (0; t, X 0 (t; t, x)))
= V0 (X 0 (0; t, x)) = V 0 (t, x) = φ(t, x) = φ(t, X 0 (t; t, x)) = φ(t),
where the passage from the first to the second line is due to the fact that the points (t, X 0 (t; t, x)) and (t, x) are on
the same characteristics. Finally, this implies
∂t φt=t ≤ 0,
which directly gives (6.11).
Step 4: Proof of the error estimate (1.13).
The solution V ǫ of (1.10) can be written (in analogue with (6.5) and (6.6)) as
ǫ
where the characteristics
ǫ
X1
V ǫ (t, x) = V0 (X 1 (t), x2 − t)
satisfies:




ǫ
∂τ X 1
ǫ
=f


 X ǫ (0) = x .
1
1
X 1 x2 − τ
ǫ
,
, X 1 , x2 − τ
ǫ
ǫ
16
(6.13)
!
(6.14)
ǫ
0
We apply Theorem 1.5, namely inequality (1.7), with uǫ and u0 replaced by X 1 and X 1 respectivly, we obtain:
cT
0
ǫ
for T ≥ Cǫ| log ǫ| with C > 0, ǫ > 0.
(6.15)
≤
X 1 − X 1 ∞
| log ǫ|
L (0,T )
Using (6.5), (6.13) and (6.15), we compute for (t, x) ∈ (0, T ) × R2 :
|V ǫ (t, x) − V 0 (t, x)|
ǫ
0
= |V0 (X 1 (t), x2 − t) − V0 (X 1 (t), x2 − t)|
ǫ
0
≤ Lip(V0 )|X 1 (t) − X 1 (t)|
cT
,
≤ Lip(V0 )
| log ǫ|
and inequality (1.13) directly follows.
2
Proof of Theorem 1.9. Application of Proposition 2.1, and same proof as Theorem 1.8.
7
Appendix: proof of Examples 1.3 and 1.6
Proof of Example 1.3. The function f can be expressed as (see for instance [23]):
−1
dv
.
f (u, t) =
0 −u + | sin 2πv|
Z 1
Z
dv
dv
Let a = −u > 0, it is easy to check that
=2
. Take
a
+
|
sin
2πv|
a
+
|
sin
2πv|
0
|v|≤1/4
Z
Z
dv
dv
Ia =
and I a,R =
.
|v|≤1/4 a + | sin 2πv|
|v|≤Ra a + 2π|v|
Z
1
We are interested in the limit a → 0 and R → ∞ with Ra → 0. We compute
I a − I a,R =
z
Z
A
Ra≤|v|≤1/4
where we have:
with c =
1
4π ,
}|
{
dv
+
a + | sin 2πv|
B
z
Z
|v|≤Ra
}|
{
2π|v| − | sin 2πv|
dv,
(a + 2π|v|)(a + | sin 2πv|)

 B → 0 as Ra → 0
p
1
c
1
A ≤
∼
∼ c | log a|,
2 sin(2πRa)
Ra
and we have chosen
Now, let v̄ = av , we also compute:
Z
I a,R = 2
0≤v̄≤R
1
R= p
.
a | log a|
dv̄
1
1
1
= (log(1 + 2πR)) ∼ log R ∼ | log a|.
1 + 2πv̄
π
π
π
Since A ≪ I a,R , this shows that I a ∼ I a,R and then
f (u, t) ∼
π
2| log |u||
which justifies (1.5).
as u → 0− ,
2
17
Proof of Example 1.6. Since f (v, τ, u, t) = g(v + τ ) − 1, then the function v ǫ defined by:
v ǫ (t) = uǫ (t) + t
satisfies

ǫ

 v ǫ = g v , t > 0,
t
ǫ

 v ǫ (0) = 0.
From the particular expression (1.8) of g with g 12 = 0, we can check that 0 ≤
implies that
v ǫ → 0 in L∞ ,
(7.1)
v ǫ (t)
ǫ
≤
1
2
for every t ≥ 0, which
then
uǫ → u0 in L∞
with u0 (t) = −t,
where
u0t = f (u0 , t) = −1.
Moreover, equation (7.1) can be written:
vtǫ =
1 vǫ
−
2
ǫ
with v ǫ (0) = 0,
therefore (solving the above equation) we get for t = δǫ| log ǫ|
uǫ (t) − u0 (t) = v ǫ (t)
=
=
=
t
ǫ
(1 − e− ǫ )
2
ǫ
(1 − e−δ| log ǫ| )
2
ǫ
1
ǫ
t
− ǫ1+δ ∼ ∼
,
2 2
2
2δ| log ǫ|
which terminates the proof of Example 1.6.
2
Acknowledgements. This work was supported by the contract ANR MICA (2006-2009). The second author
would like to thank B. Corbin for fruitful discussions.
References
[1] O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result, Arch. Ration. Mech. Anal., 170 (2003), pp. 17–61.
[2] Y. Amirat, K. Hamdache, and A. Ziani, On homogenization of ordinary differential equations and linear
transport equations, in Calculus of variations, homogenization and continuum mechanics (Marseille, 1993),
vol. 18 of Ser. Adv. Math. Appl. Sci., World Sci. Publ., River Edge, NJ, 1994, pp. 29–50.
[3] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, vol. 17 of Mathématiques & Applications
(Berlin) [Mathematics & Applications], Springer-Verlag, Paris, 1994.
[4] G. Barles and P. E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations,
SIAM J. Math. Anal., 31 (2000), pp. 925–939 (electronic).
[5] G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to
quasi-linear parabolic equations, SIAM J. Math. Anal., 32 (2001), pp. 1311–1323 (electronic).
[6] A. Benchérif-Madani and É. Pardoux, Homogenization of a semilinear parabolic PDE with locally periodic
coefficients: a probabilistic approach, ESAIM Probab. Stat., 11 (2007), pp. 385–411 (electronic).
18
[7] F. Camilli, I. Capuzzo Dolcetta, and D. A. Gomes, Error estimates for the approximation of the
effective Hamiltonian, Appl. Math. Optim., 57 (2008), pp. 30–57.
[8] I. Capuzzo-Dolcetta and H. Ishii, On the rate of convergence in homogenization of Hamilton-Jacobi
equations, Indiana Univ. Math. J., 50 (2001), pp. 1113–1129.
[9] A.-L. Dalibard, Homogenization of linear transport equations in a stationary ergodic setting, Comm. Partial
Differential Equations, 33 (2008), pp. 881–921.
[10] W. E, Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math., 45 (1992),
pp. 301–326.
[11] L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc.
Edinburgh Sect. A, 120 (1992), pp. 245–265.
[12] N. Forcadel, C. Imbert, and R. Monneau, Homogenization of fully overdamped Frenkel-Kontorova models, to appear in J. of differential equations.
[13] G. Gaudron and É. Pardoux, EDSR, convergence en loi et homogénéisation d’EDP paraboliques semilinéaires, Ann. Inst. H. Poincaré Probab. Statist., 37 (2001), pp. 1–42.
[14] T. Hou and X. Xin, Homogenization of linear transport equations with oscillatory vector fields, SIAM J.
Appl. Math., 52 (1992), pp. 34–45.
[15] C. Imbert and R. Monneau, Homogenization of first-order equations with (u/ǫ)-periodic Hamiltonians. I.
Local equations, Arch. Ration. Mech. Anal., 187 (2008), pp. 49–89.
[16] C. Imbert, R. Monneau, and E. Rouy, Homogenization of first order equations with (u/ǫ)-periodic Hamiltonians. II. Application to dislocations dynamics, Comm. Partial Differential Equations, 33 (2008), pp. 479–516.
[17] P. Lions, G. Papanicolaou, and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations,
unpublished.
[18] P.-L. Lions and P. E. Souganidis, Homogenization of “viscous” Hamilton-Jacobi equations in stationary
ergodic media, Comm. Partial Differential Equations, 30 (2005), pp. 335–375.
[19] G. Menon, Gradient systems with wiggly energies and related averaging problems, Arch. Ration. Mech. Anal.,
162 (2002), pp. 193–246.
[20] É. Pardoux, Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients:
a probabilistic approach, J. Funct. Anal., 167 (1999), pp. 498–520.
[21] R. Peirone, Homogenization of ordinary and linear transport equations, Differential Integral Equations, 9
(1996), pp. 323–334.
[22] M. Petrini, Homogenization of a linear transport equation with time depending coefficient, Rend. Sem. Mat.
Univ. Padova, 101 (1999), pp. 191–207.
[23] L. Piccinini, Homogeneization problems for ordinary differential equations, Rend. Circ. Mat. Palermo (2), 27
(1978), pp. 95–112.
[24] P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot.
Anal., 20 (1999), pp. 1–11.
[25] L. Tartar, Nonlocal effects induced by homogenization, in Partial differential equations and the calculus of
variations, Vol. II, vol. 2 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA,
1989, pp. 925–938.
[26] T. Tassa, Homogenization of two-dimensional linear flows with integral invariance, SIAM J. Appl. Math., 57
(1997), pp. 1390–1405.
19